My question for you is: Isn't the school a government thing?....
Out of curiosity, I actually did some further research on the matter. The simple answer is, "technically, public school teachers are public employees. Realistically, they are not".
On the pro-government side, teachers are paid by the school district, which gets paid by the state, thus making them employees of the government.
A report by the Center of Budget and Political Priorities marked school teachers and admins as public sector employees (thus, government employees).
However, various other reports indicate that teachers are definitely not considered government employees by the government either. They receive no state benefits such as state medical insurance nor a retirement fund. They also do not go through the government employment screening process, nor are they granted a government worker ID upon employment.
So are schools a government function? Technically yes, practically no.
At least the answer for private schools is easy: nope!
Fun fact, you are technically giving up some of your rights when you attend school. Freedom of speech, expression, and the right to bear arms being the ones that immediately come to mind. Do I think this is a bad thing? No. I'm willing to temporarily lose a few rights to get a good education.
True fact: police and campus officials on campus don't need, "probably cause" to conduct a search of your school property, just "reasonable grounds". Police searching your house for marijuana because you wore a Cheech and Chong shirt in public would be illegal; police searching your locker for marijuana for the same reason is totally within their authority.